Problem: Simplify; express your answer in exponential form. Assume $p\neq 0, z\neq 0$. $\dfrac{{(p^{5}z^{-2})^{-1}}}{{(p^{-5}z)^{4}}}$
Answer: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(p^{5}z^{-2})^{-1} = (p^{5})^{-1}(z^{-2})^{-1}}$ On the left, we have ${p^{5}}$ to the exponent ${-1}$ . Now ${5 \times -1 = -5}$ , so ${(p^{5})^{-1} = p^{-5}}$ Apply the ideas above to simplify the equation. $\dfrac{{(p^{5}z^{-2})^{-1}}}{{(p^{-5}z)^{4}}} = \dfrac{{p^{-5}z^{2}}}{{p^{-20}z^{4}}}$ Break up the equation by variable and simplify. $\dfrac{{p^{-5}z^{2}}}{{p^{-20}z^{4}}} = \dfrac{{p^{-5}}}{{p^{-20}}} \cdot \dfrac{{z^{2}}}{{z^{4}}} = p^{{-5} - {(-20)}} \cdot z^{{2} - {4}} = p^{15}z^{-2}$